Power Consumption

The power consumption of a car can be found using the following equation:

(1)
\begin{align} P = 4\[[{1\over2}] \rho Av^{3} \right] \end{align}
(2)
\begin{align} = 2 \rho Av^{3} \end{align}

ρ is the density of the air being displaced as the car travels. This density can be calculated as follows assuming an ideal gas scenario:

(3)
\begin{align} \rho = {m \over V}= {M P \over RT}\right] \end{align}

where $M$ is the molar mass in kg, leading (for $0^{\circ}C$) to

(4)
\begin{split} \rho &= 101 325 Pa/(287.058 J kg^{-1} K^{-1} \times 273.15 K) \\ & = 1.3kg/m^{3} \end{split}

(The value $287.058 J kg^{-1} K^{-1}$ in the above expression is the gas constant for dry air.)

The value of A is the drag area of each car. It is a product of the drag coefficient and the frontal area of the car i.e. $A = c_{d} A_{car}$. In order to calculate the power consumed by different vehicles I assumed that the car would be driven on a highway therefore no power would be consumed in braking and accelerating. The speed assumed was 24.6m/s (55mph). The following table shows the power consumption for different vehicles:

Vehicle Drag Area (m2) Power Consumed(kW)
Hummer (2003) 2.44 94.4
Hummer H1 (1993) 1.68 65.0
Land Rover 1.6 61.9
Hummer H3 (2006) 1.56 60.4
Chevrolet Tahoe (Hybrid SUV - 2008) 1.4 54.6
Lexus RX (SUV) 0.36 42.2
Ford Escape Hybrid (2005) 1.08 41.8
Volvo 740 0.81 31.4
Chevrolet Cavalier (2003) 0.79 30.6
Honda Civic 0.68 26.3
SSC Aero (2003) 0.66 25.5
VW Polo GTi 0.65 25.2
Toyota Prius (2004) 0.58 22.4
Mercedes Bionic Car 0.55 21.3
Mercedes-Benz E-class Coupe (2010) 0.51 19.7
Honda Insight 0.47 18.2
Twike 0.23 8.9
Aptera 2 series (2009) 0.19 7.35

These data are important because they give an idea of how much power each car requires to travel the same distance. It means that if a 2003 Hummer travels 55 miles in one hour (55mph = 24.6m/s) it uses 94.4kWh of energy whereas an Aptera 2 series would only require 7.35kWh. The power consumed is dependent on the frontal area of the car and therefore the smaller that area the less the energy that the car uses when moving. Some car-makers are now designing cars that are narrower with less aerodynamic drag which is the case with the Aptera 2 series which requires the least amount of energy in comparison to the other cars in the table. The power consumed is also dependent on the speed with which the car is driving. For example if the 2003 Hummer travels at 49.2m/s (110mph) in half an hour, then the power it consumes will be 377.75kW. Therefore to travel the same distance would require 4 times more energy than if it were traveling at 24.6m/s. It can be concluded that a narrower, slower car consumes less energy.

### References

http://wikipedia.org/

http://www.withouthotair.com/

page revision: 55, last edited: 09 Sep 2009 01:13